Question

the circle below has center o, and its radius is 5 in. given that m∠aob = 50°, find the length of the major arc \\(\overarc{acb}\\). give an exact answer in terms of π, and be sure to include the correct unit in your answer. length of major arc \\(\overarc{acb}\\) :

Explanation:

Step1: Find the measure of the major arc's central angle

The total degrees in a circle is \( 360^\circ \). The central angle for the minor arc \( \overset{\frown}{AB} \) is \( 50^\circ \), so the central angle for the major arc \( \overset{\frown}{ACB} \) is \( 360^\circ - 50^\circ = 310^\circ \).

Step2: Recall the arc length formula

The formula for the length of an arc is \( L=\frac{\theta}{360^\circ}\times 2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step3: Substitute the values into the formula

Here, \( \theta = 310^\circ \) and \( r = 5 \) in. Plugging these into the formula:
\( L=\frac{310^\circ}{360^\circ}\times 2\pi\times 5 \)
Simplify the fraction \( \frac{310}{360}=\frac{31}{36} \), and \( 2\times5 = 10 \), so:
\( L=\frac{31}{36}\times 10\pi=\frac{310\pi}{36}=\frac{155\pi}{18} \)

Answer:

\(\frac{155}{18}\pi\) inches